Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras of Type A

0.23 MB·English

by  Ayumu Hoshino

Polyhedral Realizations of Crystal Bases for
Modified Quantum Algebras of Type A
4
0 Hoshino Ayumu ∗ Nakashima Toshiki †
0
2 Department of Mathematics, Department of Mathematics,

n Sophia University Sophia University,
a
J Tokyo 102-8554, Japan Tokyo 102-8554, Japan

9

]
A
Q
Abstract
.
h
t We describe the crystal bases of modified quantum algebras and its connected com-
a
ponent containing “zero vector” by the polyhedral realization method for the type A
m n
and A(1). We also present the explicit form of the unique highest weight vector in the
[ 1

connected component.
1
v
7
8
0 1 Introduction
1
0
4 Let U (g) :=< e ,f ,qh > (I = 1,2,··· ,n) be the quantum algebra associated with the
q i i i∈I
0
symmetrizable Kac-Moody Lie algebra g and U−(g) :=< f > (resp. U+(g) :=< e > ) be
/ q i i∈I q i i∈I
h the subalgebra of U (g). Kashiwara showed that the subalgebras U±(g) has a unique crystal
t q q
a base(L(∓∞),B(∓∞))andarbitraryintegrablehighest weightU (g)-moduleV(λ)hasaunique
m q
crystal baase (L(λ),B(λ)) ([2]).
:
v He also proved that a tensor product of crystal bases for modules is again a crystal base
i
X for tensor product of corresponding modules, which is one of the most beautiful and usefull
r properties in theory of crystal bases.
a
The term “crystal ” implies a combinatorial notion abstracting the properties of crystal
base without assuming existence of the corresponding modules. We shall see its examples in
Sect.2.1. Wecandefinetensorproductstructureoncrystalsinasimilarmannertocrystalbases.
Indeed, some crystals (and their tensor products) are used to realize crystal base B(∞)([8])
and B(λ)([7]) by the polyhedral realization method. Polyhedral realization of crystal bases is
the method embedding crystal bases in some infinite-dimensional vector space and describing
its image as a set of lattice points in certain convex polyhedron([7],[8]).
ThemodifiedquantumalgebraU˜ (g) := U (g)a (resp. U (g)a := U (g)/ U (g)(qh−
q λ∈P q λ q λ q q
qhhi,λi)) is given by modifiying the Cartan part of U (g). Lusztig showed that it has a crystal
q
base (L(U˜ (g)),B(U˜ (g))) = (⊕ L(U (g)a L),⊕ B(U (g)a )) ([6]) and Kashiwara dPescribe its
q q λ q λ λ q λ
∗e-mail: a-hoshin@hoffman.cc.sophia.ac.jp
†e-mail: [email protected]
1
important properties ([5]). One of them is that the existence of the following isomorphism of
crystals:
B(U (g)a ) ∼= B(∞)⊗T ⊗B(−∞)
q λ λ
where T is the crystal given in Sect.2.2. We have already the polyhedral realization of B(±)
λ
and then we can get the polyhedral realization of B(U (g)a ) (Sect.4), which is, in general, not
q λ
connected, on the contrary, including infinitely many components. So, next we try to describe
some specific connected component B (λ) containing u ⊗t ⊗u and the explicit form of
0 ∞ λ −∞
the unique highest weight vector in B (λ) by the polyhedral realization method under certain
0
assumptioin on the weight λ in the cases g = A and A(1).
n 1
The organization of this paper is as follows: in Sect.2., we review the theory of crystal
bases and crytals. We also prepare several ingredients to use in the subsequent sections, e.g.,
crystals B , T and explain polyhedral realization of B(±∞). In Sect.3., we define the modified
i λ
˜
quantum algebra U (g) and see its crystal base andin Sect.4., itspolyhedral realizationis given.
q
InSect.5.,we consider themodified quantum algebraof typeA . Wedescribe thehighest weight
n
vector inthe connected component B (λ) including u ⊗t ⊗u and the polyhedal realization
0 ∞ λ −∞
(1)
of B (λ) under certain condition on λ. In the last section, we treat the affine A -case. In this
0 1
case, we succeeded in presenting the explicit form of the connected component of B (λ) and
0
the unique highest weight vector in B (λ) for a poisitive level weight λ.
0
2 Crystal Base and Crystals
2.1 Definition of Crystal Base and Crystal
In this section, we shall review crystal bases and crystals. We follow the notations and termi-
nologies to [7][8].
We fix a finite index set I. Let A = (a ) be a generalized symmetrizable Cartan
ij i,j∈I
matrix, (t,{α } ,{h } ) be the associated Cartan data and g be the associated Kac-Moody
i i∈I i i∈I
Lie algebra where α (resp. h ) is called an simple root (resp. simple coroot). Let P be a weight
i i
lattice, P∗ be a dual lattice including {h } and Q := Q(q)α be a root lattice. Let
i ∈I i∈I i
Uq(g) := hqh,ei,fiii∈I,h∈P∗ be the quantum algebra defined by the usual relations, which is an
L
associative and Hopf algebra over the field Q(q). (We do not write down the Hopf algebra
structure here.)
Now we give the definition of crystal base. Let O be the category whose objects are
int
U (g)-module that it has a weight space decomposition and for any element u, there exists
q
positive integer l such that e ···e u = 0 for any k > l. It is well-known that the category
i1 ik
O is a semisimple category and all simple ojbects are parametrized by the set of dominant
int
integral weights P . Let M be a U (g)-module in O . For any u ∈ M (λ ∈ P), we have the
+ q int λ
unique expression:
(n)
u = f u ,
i n
n≥0
X
˜
where u ∈ Kere ∩M . By using this, we define the Kashiwara operators e˜,f ∈ End(M)
n i λ+nαi i i
(i ∈ I),
(n−1) ˜ (n+1)
e˜u := f u , f u := f u .
i i n i i n
n≥1 n≥0
X X
2
here note that we can define the Kashiwara operators e˜,f˜ ∈ End(U±(g)) by the similar
i i q
manner([2]). Let A ⊂ Q(q) be the subring of rational functions which are regular at q = 0. Let
M be a U (g)-module in O (resp. U±(g)).
q int q
Definition 2.1 ([2]). A pair (L,B) is a crystal base of M (resp. U±(g)), if it satisfies:
q
(i) LisfreeA-submoduleofM (resp. U±(g))andM ∼= Q(q)⊗ L(resp. U±(g) ∼= Q(q)⊗ L).
q A q A
(ii) B is a basis of the Q-vector space L/qL.
(iii) L = ⊕ L , B = ⊔ B where L := L∩M , B := B ∩L /qL.
λ∈P λ λ∈P λ λ λ λ λ
˜
(iv) e˜L ⊂ L and f L ⊂ L.
i i
˜ ˜ ˜
(v) e˜B ⊂ B ⊔{0} and f B ⊂ B ⊔{0} (resp. f B ⊂ B) ( e˜ and f acts on L/qL by (iv)).
i i i i i
˜
(vi) For u,v ∈ B, f u = v ⇐⇒ e˜v = u.
i i
The unit of subalgebra U∓(g) is denoted by u . We set
q ±∞
˜ ˜
L(∞) := Af ···f u ,
il i1 ∞
ij∈XI,l≥0
L(−∞) := Ae˜ ···e˜ u ,
il i1 −∞
ij∈XI,l≥0
˜ ˜
B(∞) := {f ···f u mod qL(∞)|i ∈ I,l ≥ 0},
il i1 ∞ j
B(−∞) := {e˜ ···e˜ u mod qL(−∞)|i ∈ I,l ≥ 0}.
il i1 −∞ j
Theorem 2.2 ([2]). A pair (L(±∞),B(±∞)) is a crystal base of U∓(g).
q
Now we introduce the notion crystal, which is obtained by abstracting the combinatorial
properties of crystal bases.
Definition 2.3. A crystal B is a set endowed with the following maps:
wt : B −→ P,
ε : B −→ Z⊔{−∞}, ϕ : B −→ Z⊔{−∞} for i ∈ I,
i i
˜
e˜ : B ⊔{0} −→ B ⊔{0}, f : B ⊔{0} −→ B ⊔{0} for i ∈ I,
i i
˜
e˜(0) = f (0) = 0.
i i
those maps satisfy the following axioms: for all b,b ,b ∈ B, we have
1 2
ϕ (b) = ε (b)+hh ,wt(b)i,
i i i
wt(e˜b) = wt(b)+α if e˜b ∈ B,
i i i
˜ ˜
wt(f b) = wt(b)−α if f b ∈ B,
i i i
˜
e˜b = b ⇐⇒ f b = b (b ,b ∈ B),
i 2 1 i 1 2 1 2
˜
ε (b) = −∞ =⇒ e˜b = f b = 0.
i i i
3
Indeed, if (L,B) is a crystal base, then B is a crystal.
Definition 2.4. (i) Let B and B be crystals. A strict morphism of crystals ψ : B −→ B
1 2 1 2
is a map ψ : B ⊔{0} −→ B ⊔{0} satisfying the following: (1) ψ(0) = 0. (2)If b ∈ B
1 2 1
and ψ(b) ∈ B , then
2
wt(ψ(b)) = wt(b), ε (ψ(b)) = ε (b), ϕ (ψ(b)) = ϕ (b).
i i i i
˜
and the map ψ commutes with all e˜ and f .
i i
(ii) An injective strict morphism is called an embedding of crystals. We call B is a subcrystal
1
of B , if B is a subset of B and becomes a crystal itself by restricting the data on it
2 1 2
from B .
2
The following examples will play an important role in the subsequent sections.
Example 2.5. Let T := {t } (λ ∈ P) be the crystal consisting of one element t defined by
λ λ λ
˜
wt(t ) = λ, ε (t ) = ϕ (t ) = −∞ , e˜(t ) = f (t ) = 0.
λ i λ i λ i λ i λ
Example 2.6. For i ∈ I, the crystal B := {(x) : x ∈ Z} is defined by:
i i
wt((x) ) = xα , ε ((x) ) = −x, ϕ ((x) ) = x,
i i i i i i
ε ((x) ) = −∞, ϕ ((x) ) = −∞ for j 6= i,
j i j i
˜
e˜ (x) = δ (x+1) , f (x) = δ (x−1) .
j i i,j i j i i,j i
Note that as a set B is identidied with the set of integers Z.
i
2.2 Polyhedral Realization of B(±∞)
We review the polyhedral realization of the crystal B(±∞) following to [8].
We consider the following additive groups:
Z+∞ := {(··· ,x ,··· ,x ,x ) : x ∈ Z and x = 0 for k ≫ 0},
k 2 1 k k
Z−∞ := {(x ,x ,··· ,x ,···) : x ∈ Z and x = 0 for k ≫ 0}.
−1 −2 −k −k −k
We will denote by Z+∞ ⊂ Z+∞ ( resp. Z−∞ ⊂ Z−∞ ) the semigroup of non-negative (resp.
≥0 ≤0
non-positive) integer sequences. Take an infinite sequence of indices ι+ = (··· ,i ,··· ,i ,i )
k 2 1
(resp. ι− = (i ,i ,··· ,i ,···)) from I such that
−1 −2 −k
i 6= i for any k, and ♯{k > 0 (resp. k < 0) : i = i} = ∞ for any i ∈ I. (2.1)
k k+1 k
We can associate to ι+ ( resp. ι−) a crystal structure on Z+∞ (resp. Z−∞) (see [8]) and denote
it by Z+∞ (resp. Z−∞). Let B be the crystal given in Example2.6. We obtain the following
ι+ ι− i
embeddings([3]):
Ψ+ : B(∞) ֒→ B(∞)⊗B (u 7→ u ⊗(0) ),
i i ∞ ∞ i
Ψ− : B(−∞) ֒→ B ⊗B(−∞) (u 7→ (0) ⊗u ).
i i −∞ i −∞
4
Iterating Ψ+ (resp. Ψ−) according to ι+ (resp. ι−), we get the Kashiwara embedding([3]);
i i
Ψ : B(∞) ֒→ Z+∞ ⊂ Z+∞ (u 7→ (··· ,0,··· ,0,0,0)), (2.2)
ι+ ≥0 ι+ ∞
Ψι− : B(−∞) ֒→ Z−≤∞0 ⊂ Z−ι−∞ (u−∞ 7→ (0,0,0,··· ,0,···)). (2.3)
We consider the following infinite dimensional vector spaces and their dual spaces:
Q+∞ := Q⊗ Z+∞ = {~x = (··· ,x ,··· ,x ,x ) : x ∈ Q and x = 0 for k ≫ 0},
Z k 2 1 k k
Q−∞ := Q⊗ Z−∞ = {~x = (x ,x ,··· ,x ,···) : x ∈ Q and x = 0 for k ≫ 0},
Z −1 −2 −k −k −k
(Q±∞)∗ := Hom(Q±∞,Q).
We will write a linear form ϕ ∈ (Q+∞)∗ as ϕ(~x) = ϕ x (ϕ ∈ Q). Similarly, we write
k≥1 k k j
ϕ ∈ (Q−∞)∗ as ϕ(~x) = ϕ x (ϕ ∈ Q).
k≤−1 k k j P
For the sequence ι+ = (i ) (resp. ι− = (i ) ) and k ≥ 1 (resp. k ≤ −1), we set
k k≥1 k k≤−1
P
k(+) := min{l : l > k > 0 (resp.0 > l > k) and i = i },
k l
if it exists, and
k(−) := max{l : 0 < l < k (resp.l < k < 0) and i = i },
k l
if it exists, otherwise k(+) = k(−) = 0.
We define a linear form β (k ≥ 0) on Q+∞ by
k
x + hh ,α ix +x (k ≥ 1),
β (~x) := k k<j<k(+) ik ij j k(+) (2.4)
k
(0 (k = 0).
P
We also define a linear form β (k ≤ 0) on Q−∞ by
k
y + hh ,α iy +y (k ≤ −1),
β (~y) := k(−) k(−)<j<k ik ij j k (2.5)
k
(0 (k = 0).
P
By using these linear forms, let us define a piecewise-linear operator S = S on (Q±∞)∗ as
k k,ι
follows:
ϕ−ϕ β if ϕ > 0,
k k k
S (ϕ) := (2.6)
k
(ϕ−ϕkβk(−) if ϕk ≤ 0,
for ϕ(~x) = ϕ x ∈ (Q±∞)∗. Here we set
k k
P Ξι± := {S±jl···S±j2S±j1(±xj0)|l ≥ 0,j0,j1,··· ,jl ≥ 1},
Σι± := {~x ∈ Z±∞ ⊂ Q±∞|ϕ(~x) ≥ 0 for any ϕ ∈ Ξι±},
We impose on ι+ and ι− the following assumptions (P),(N):
(P) for ι+, if a positive k satisfies k(−) = 0 then ϕ ≥ 0 for any ϕ(~x) = ϕ x ∈ Ξ ,
k k k k ι+
(N) for ι−, if a negative k satisfies k(+) = 0 then ϕk ≤ 0 for any ϕ(~x) =Pkϕkxk ∈ Ξι−.
Theorem 2.7 ([8]). Let ι± be the indices of sequences which are satisfiedP(2.1) and the as-
sumptions (P),(N). Suppose Ψι+ : B(∞) ֒→ Z∞ι+ and Ψι− : B(−∞) ֒→ Z−ι−∞ are the Kashiwara
∼ ∼
embeddings. Then, we have Im(Ψι+)(= B(∞)) = Σι+, Im(Ψι−)(= B(−∞)) = Σι−.
5
3 Modified Quantum Algebras and its Crystal Base
We define the left U (g)-module U (g)a ([4]) by the relation: qha = qhh,λia . Then U˜ (g) =
q q λ λ λ q
⊕ U (g)a has an algebra structure by
λ∈P q λ
(i) a P = Pa for ξ ∈ Q and P ∈ U (g)
λ λ−ξ q ξ
(U (g) := {P ∈ U (g); qhPq−h = qhh,ξiP for any h ∈ P∗}).
q ξ q
(ii) a a = δ a ,
λ µ λ,µ λ
and we call this algebra modified quantum algebra.
Let M be a U (g)-module with the weight space decomposition M = ⊕ M . Then a is
q λ∈P λ λ
a projection a : M −→ M .
λ λ
In [6], it is revealed that modified quantum algebra U˜ (g) has a crystal structure and in [4]
q
its crystal base is described as follows:
Theorem 3.1 ([4]).
B(U (g)a ) = B(∞)⊗T ⊗B(−∞),
q λ λ
B(U˜ (g)) = B(∞)⊗T ⊗B(−∞).
q λ
λ∈P
M
4 Polyhedral Realization of B(U (g)a )
q λ
4.1 Crystal structure of Z∞[λ]
LetZ+∞ andZ−∞ beasinSect.2.2,Wetaketheindicessequenceι := (ι+,t ,ι−) = (··· ,i ,i ,t ,i ,i ,···)
ι+ ι− λ 2 1 λ −1 −2
and weight λ ∈ P. We set Z∞[λ] := Z∞ ⊗ T ⊗ Z−∞. The crystal structure on Z∞[λ] asso-
ι ι+ λ ι− ι
ciated with ι and λ is defined as follows: We identify Z+∞ ⊗T ⊗Z−∞ with Z∞. Therefore,
λ
Z∞[λ] is regarded as a sublattice of Q∞ = Q ⊗ Z∞. Thus, we can deonte ~x ∈ Z∞[λ] by
Z ι
~x = (··· ,x ,x ,t ,x ,x ,···). For ~x ∈ Q∞, we define a linear function σ (~x) (k ∈ Z) by:
2 1 λ −1 −2 k
x + hh ,α ix (k ≥ 1),
k j>k ik ij j
σ (~x) := −hh ,λi+x + hh ,α ix (k ≤ −1), (4.1)
k  ikP k j>k ik ij j
−∞ (k = 0).
P

Since x = 0 for j ≫ 0,σ is well-defined. Let σ(i)(~x) := max σ (~x) and
j k k:ik=i k
M(i) = M(i)(~x) := {k : i = i,σ (~x) = σ(i)(~x)}. (4.2)
k k
Note that σ(i)(~x) ≥ 0, and that M(i) = M(i)(~x) is a finite set if and only if σ(i)(~x) > 0. Now,
we define the map e˜ : Z∞[λ]⊔{0} −→ Z∞[λ]⊔{0} , f˜ : Z∞[λ]⊔ {0} −→ Z∞[λ]⊔{0} , by
i i
˜
e˜(0) = f (0) = 0 and
i i
(f˜(~x)) = x +δ if M(i) exists; otherwise f˜(~x) = 0, (4.3)
i k k k,minM(i) i
(e˜(~x)) = x −δ if M(i) exists; otherwise e˜(~x) = 0. (4.4)
i k k k,maxM(i) i
6
where δ is Kronecker’s delta. We also define the weight function and the function ε and ϕ
i,j i i
on Z∞[λ] as follows:
wt(~x) := λ− ∞ x α , ε (~x) := σ(i)(~x),
j=−∞ j ij i (4.5)
ϕ (~x) := hh ,wt(~x)i+ε (~x).
i i i
P
We denote this crystal by Z∞[λ].
ι
Since there exist the embedings of crystals: B(±) ֒→ Z±∞, we obtain
ι±
Theorem 4.1.
Ψ(λ) : B(∞)⊗T ⊗B(−∞) ֒→ Z+∞ ⊗T ⊗Z−∞(= Z∞[λ])
ι λ ι+ λ ι− ι
u ⊗t ⊗u 7→ (··· ,0,0,t ,0,0,···)
∞ λ −∞ λ
is the unique stirict embedding which is associated with ι := (··· ,i ,i ,t ,i ,i ,···).
2 1 λ −1 −2
(λ)
4.2 The image of Ψι
¯
Fix a sequence of indices ι as above. We define a linear function β (~x) as follows:
k
¯
β (~x) = σ (~x)−σ (~x) (4.6)
k k k(+)
where σ is defined by (4.1). Since hh ,α i = 2 for any i ∈ I, we have
k i i
x + hh ,α ix +x (k ≥ 1 or k(+) ≤ −1),
β¯ (~x) = k k<j<k(+) ik ij j k(+)
k
(−hhikP,λi+xk + k<j<k(+)hhik,αijixj +xk(+) (k ≤ −1 and k(+) > 0).
P ¯ ¯
Using this notation, we define an operator S = S for a linear function ϕ(~x) = c +
k k,ι
∞ ϕ x (c,ϕ ∈ Q) as follows:
−∞ k k k
P
¯
ϕ−ϕ β if ϕ > 0,
S¯ (ϕ) := k k k (4.7)
k ¯
(ϕ−ϕkβk(−) if ϕk ≤ 0.
An easy check shows (S¯ )2 = S¯ . For a sequence ι and an integral weight λ, we denote by
k k
Ξ [λ] the subset of linear forms which are obtained from the coordinate forms x , x (j ≥ 1)
ι j −j
by applying transformations S¯ . In other words, we set
k
Ξ+[λ] := {S¯ ···S¯ (x ) : l ≥ 0, j ,··· ,j > 0}
ι jl j1 j0 0 l
Ξ−[λ] := {S¯ ···S¯ (−x ) : k ≥ 0, j ,··· ,j > 0}, (4.8)
ι −jk −j1 −j0 0 k
Ξ [λ] := Ξ+[λ]∪Ξ−[λ].
ι ι ι
Now we set
Σ [λ] := {~x ∈ Z∞[λ](⊂ Q∞) : ϕ(~x) ≥ 0 for any ϕ ∈ Ξ [λ]}. (4.9)
ι ι ι
By Theorem 2.7, we have
7
Theorem 4.2. Suppose that ι± satisfys the assumption (P),(N),and (2.1).Let Ψ(λ) : B(∞)⊗
ι
T ⊗B(−∞) ֒→ Z∞[λ] be the embedding of (4.1). Then Im(Ψ(λ))(∼= B(∞)⊗T ⊗B(−∞)) is
λ ι ι λ
equal to Σ [λ].
ι
Remark. Under the assumptions (P) and (N), both Ξ+[λ] and Ξ−[λ] are closed by the
ι ι
actions of S¯ ’s, since S (k < 0) (resp. S (k > 0)) acts identically on Ξ+[λ] (resp. Ξ−[λ]).
k k k ι ι
The following lemma will be used lalter.
Lemma 4.3. Let Ξ be a set of linear functions on Q∞. Suppose that Ξ is closed by actions of
all S¯ ’s, then the set
k
Σ = {~x ∈ Z∞[λ]|ϕ(~x) ≥ 0 for any ϕ ∈ Ξ }
ι ι
is a sub-crystal of Z∞[λ].
ι
Proof. It suffices to show that Σ is closed under the actions of f˜ and e˜. For~x ∈ Σ, suppose
i i
˜
f ~x = (··· ,x +1,···). For any ϕ = c+ ϕ x ∈ Ξ (c,ϕ ∈ Q), we need to show that
i k j j j
Pϕ(f˜~x) ≥ 0. (4.10)
i
˜
Since ϕ(f (~x)) = ϕ(~x) + ϕ ≥ ϕ , it is enough to consider the case when ϕ < 0. By
i k k k
˜
definition of f ~x, we have σ < σ . This shows that
i k(−) k
¯
σ < σ ⇐⇒ β = σ −σ < 0
k(−) k k(−) k(−) k
¯
=⇒ β ≤ −1.
k(−)
Therefore, it follows that
˜
ϕ(f ~x) = ϕ(~x)+ϕ
i k
¯
≥ ϕ(~x)−ϕ β
k k(−)
= (S¯ ϕ)(~x) ≥ 0.
k
Suppose that e˜~x = (··· ,x −1,···). We need to show that
i k
ϕ(e˜~x) ≥ 0. (4.11)
i
Since ϕ(e˜(~x)) = ϕ(~x) − ϕ ≥ −ϕ , it is enough to consider the case when ϕ > 0. By
i k k k
definition of e˜~x, we have σ > σ . This shows that
i k k(+)
¯
σ > σ ⇐⇒ β = σ −σ > 0
k k(+) k k k(+)
¯
=⇒ β ≥ 1.
k(−)
Therefore, it follows that
ϕ(e˜~x) = ϕ(~x)−ϕ
i k
¯
≥ ϕ(~x)−ϕ β
k k
= (S¯ ϕ)(~x) ≥ 0.
k
8
5 Polyhedral Realization of B(U (g)a ) of Type A
q λ n
In this section, we shall describe the crystal structure of the component including t in B(∞)⊗
λ
T ⊗B(−∞) for the case of type A .
λ n
It will be convenient for us to change the indexing set for Z∞ from Z to Z ×[1,n]. We
≥1 ≥1
will do this with the help of the bijection Z ×[1,n] → Z given by ((j;i) 7→ (j −1)n+i).
≥1 ≥1
Thus, we will write an element ~x ∈ Z+∞ as doubly-indexed family (x ) of nonnegative
j;i j≥1,i∈[1,n]
integers. Simillarly, using that Z ×[1,n] → Z ((j;i) 7→ −jn+i−1) is bijective, we will
≥1 ≤−1
write an element ~x ∈ Z−∞ as doubly-indexed family (x ) of nonpositive integers.
−j;i j≥1,i∈[1,n]
Therefore, we can write that ~x ∈ Z∞ as (··· ,x ,x ,t ,x ,x ,···). We will adopt
1;2 1;1 λ −1;n −1;n−1
the convention that x = x = 0 unless i ∈ [1,n]
j;0 j;n+1
To state the main theorem, we prepare several things. For x ∈ R, set (x) := max(0,x).
+
Let λ = λ Λ be an integral weight satisfying λ ,··· ,λ > 0 and λ ,··· ,λ ≤ 0 for
1≤i≤n i i 1 i0 i0+1 n
some i and for (j;i) ∈ Z ×[1,n] set
0 ≥1
P
(−λ +(−λ +(···+(−λ ) ) ···) ) if 1 ≤ j ≤ i ≤ n,
C = −j+i+1 −j+i+2 −j+n+1 + + + +
−j;i 0 otherwise.
(cid:26)
We will use the following lemma frequently:
Lemma 5.1. For real numbers r ,··· ,r , we have,
1 n
r +(r +(r +···+(r +(r ) ) ) ) = max(r ,r +r ,··· ,r +r +···+r )
1 2 3 n−1 n + + + + 1 1 2 1 2 n
Proof.
We can easily show from the fact : r +(r ) = max(r ,r +r ) and iterating this.
1 2 + 1 1 2
By the above lemma, we can write
C = max(0,−λ ,−λ −λ ,··· ,−λ −λ −···−λ ).
−j;i −j+i+1 −j+i+1 −j+i+2 −j+i+1 −j+i+2 −j+n+1
Theorem 5.2. Let ι = (··· ,2,1,n,··· ,2,1,t ,n,n − 1,··· ,1,n,n − 1,···) be an infinite
λ
sequence and λ and C be as above. We define
−j;i
′ ¯ ¯
Ξ [λ] := {S ···S (x +C ) : k ≥ 0, i ∈ I,j ≥ 1, j ,··· ,j ≥ 1},
ι −jk −j1 −j;i −j;i 1 k
Σ′[λ] := {~x ∈ Z∞[λ](⊂ Q∞) : ϕ(~x) ≥ 0 for any ϕ ∈ Ξ′[λ]}
ι ι ι
and denote the connected component of Im(Ψ(λ)) containing ~0 := (··· ,0,0,t ,0,0,···) by
ι λ
B (λ). Then we have
0
(i) B (λ) = Σ [λ]∩Σ′[λ].
0 ι ι
(ii) Let v be the unique highest weight vector in B (λ). Then we have
λ 0
v = (··· ,0,0,t ,−C ,−C ,··· ,−C ,···),
λ λ −1;n −1;n−1 −j;i
9
Proof. Since Ξ′[λ] is closed under the actions of S¯ ’s, by Lemma 4.3 Σ [λ] ∩ Σ′[λ] has a
ι k ι ι
crystal structure unless it is empty. We will show that Σ [λ]∩Σ′[λ] contains ~0, which implies
ι ι
that Σ [λ]∩Σ′[λ] is non-empty, and has the unique highest weight vector. First, we will show
ι ι
that
x = (··· ,0,0,t ,−C ,−C ,··· ,−C ,···) (5.1)
0 λ −1;n −1;n−1 −j;i
isahighestweightvectorinZ∞[λ]. Wesety := −λ +(−λ +(···+(−λ ) ) ) .
ι −j;i −j+i+1 −j+i+2 −j+n+1 + + +
Thus, we have C = −(y ) . Due to (4.4), it suffices to show
−j;i −j;i +
σ (x ) ≤ 0 for j ≥ 1, i ∈ I.
−j:i 0
(In the case j < 0, trivially σ (x ) = 0.) We consider the following four cases:
−j:i 0
(I) j = 1. (II) i = n. (III) 1 ≤ i < j ≤ n. (IV) 1 < j ≤ i < n.
(I) The case j = 1.
We will show that σ (x ),σ (x ),··· ,σ (x ) ≤ 0. Note the following simple fact:
−1,n 0 −1;n−1 0 −1;1 0
−(−a) −a ≤ 0, for any a ∈ R. (5.2)
+
We can write y := −λ +(−λ +(···+(−λ ) ) ···) . By the definition of σ ,
−1;i+1 i+1 i+2 n + + + −j:i
we have σ (x ) = −(−λ +(y ) ) +(y ) −λ . By (5.2), we obtain σ (x ) ≤ 0.
−1;i 0 i −1;i+1 + + −1;i+1 + i −1;i 0
This shows σ (x ),σ (x ),··· ,σ (x ) ≤ 0.
−1;n 0 −1;n−1 0 −1;1 0
(II) The case i = n.
We shall show σ (x ) ≤ 0 (1 ≤ ∀j ≤ n) by the induction on j. If j = 1, it is true by (I).
−j;n 0
Suppose j > 1. We can write σ (x ) = −(−λ ) + (−λ + (−λ ) ) −
−j;n 0 −j+n+1 + −j+n+1 −j+n+2 + +
(−λ ) + σ (x ). Now, set A := σ (x ) − σ (x ). Since σ (x ) ≤ 0 by
−j+n+2 + −j+1;n 0 −j;n 0 −j+1;n 0 −j+1;n 0
the induction hypothesis, it is sufficient to show A ≤ 0.
If λ , λ ≥ 0 λ , λ ≤ 0, then obviously A = 0. If λ ≥ 0
−j+n+1 −j+n+2 −j+n+1 −j+n+2 −j+n+1
and λ ≤ 0, we can write A = (−λ − λ ) + λ . In this case, if
−j+n+2 −j+n+1 −j+n+2 + −j+n+2
−λ − λ ≤ 0, then A = λ ≤ 0. If −λ − λ ≥ 0, then A =
−j+n+1 −j+n+2 −j+n+2 −j+n+1 −j+n+2
−λ ≤ 0. If λ ≤ 0 and λ ≥ 0, obviously A = 0.
−j+n+1 −j+n+1 −j+n+2
(III) The case 1 ≤ i < j ≤ n.
By the definition, C = 0 for i < j. We can write
−j;i
σ (x ) = C −C +σ (x ).
−j;i 0 −i−1:i+1 −i:i −i;i 0
By Lemma 5.1, we have
C = max(0,−λ ,−λ −λ ,··· ,−λ −λ −···−λ ),
−i:i 1 1 2 1 2 −i+n+1
C = max(0,−λ ,−λ −λ ,··· ,−λ −λ −···−λ ).
−i−1:i+1 1 1 2 1 2 −i+n
Therefore, we obtain C ≥ C . This shows σ (x ) ≤ 0.
−i:i −i−1:i+1 −j;i 0
(IV) The case 1 < j ≤ i < n.
We will show σ (x ) ≤ 0 by the induction on 1 < j ≤ i < n. We can write:
−j;i 0
σ (x ) = −C +C +C −C +σ (x )
−j;i 0 −j;i −j;i+1 −j+1;i−1 −j+1;i −j+1;i 0
10